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What are the necessary conditions (not saying sufficient conditions) in mathematical terms that a deterministic dynamic system can transit to deterministic chaos?

We collected yet:

  1. A positive feedback loop
  2. Non-linearity
  3. Minimum of three instable eigenmodes
  4. ...???
Abhimanyu Pallavi Sudhir
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al-Hwarizmi
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2 Answers2

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The dimension should be 3 or larger. If the dimension is smaller then 3 the existence and uniqueness theorem for differential equations will tell you that functions can't intersect (since you want them to be continuous and differentiable).

In 1 dimension this means you can only have movement in one direction

In 2 dimensions this means that your value either goes to infinity or to a perticular point.

You need 3 dimensions or more to get these strange attractors and that weird chaotic behaviour.

Nick
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    This is true only for continuous-time dynamical systems. In discrete time, even one-dimensional systems (such as the logistic map) can behave chaotically. Also, even in continuous time, two-dimensional systems can still exhibit (non-chaotic) limit cycles in addition to divergence and point attractors. – Ilmari Karonen Jun 23 '13 at 22:10
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    Yes of course, the theorem I used to give the condition was also one for differential equations of continuous variables. My bad :(. – Nick Jun 23 '13 at 23:28
  • how would time dilatation effect? and how the degree of memory? – al-Hwarizmi Jun 24 '13 at 09:37
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    I think if you're thinking about relativity the best way to parametrize is to use the proper time since it's invariant for all observers.

    And considering memory-effects, I believe they are more statistical which breaks down your "deterministic" in deterministic chaos.

     – Nick Jun 24 '13 at 10:26
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    Also, the chaotic manifold doesn't need to be an attractor, it can aslo be a chaotic saddle (leading to transient chaos). – Xurtio Jul 09 '13 at 15:04
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According to Nonlinear Dynamics and Chaos by Steven Strogatz The requirements for chaos are:

  1. Deterministic system (only one future for each state)

  2. Irregular spatial, temporal, or spatiotemporal patterns (a qualitative feature)

  3. A positive maximum Lyapunov exponent.

3) is pretty much the quantitative standard in journals of chaos, assuming you meet the conditions of 1). 2) is subjective and there's things like "stable chaos" and there can be periodic behavior that appears irregular but just has a really long period before it repeats itself, so you have to be careful with 2).

Xurtio
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